If the Captain has 3 War Ships and 1 of those is Maxed Materials, and he/she logs out, logs in re-joins crew gets another one of his/her ships out and then everyone on his/her first ship then teleports to them and starts a new mat run do they get double Materials? (Note: the first ship hasn't Docked and still has full health).

Even then, the Captain can't launch another ship until he boards the first ship and ports. Exception is when your ship is stuck.

I'd have to disagree with you there, I took my War Frigate out and had someone to tp to my ship, I told them not to dock it as I wanted to try something, I logged out then back in and was able to launch my War Brig and sail next to my War Frigate.